An instance E of the data type Explorer is a decorator to explore the structure of the plane map underlying the Nef polyhedron. It inherits all topological adjacency exploration operations from Topological_explorer. Explorer additionally allows one to explore the geometric embedding.

The position of each vertex is given by a so-called extended point, which is either a standard affine point or the tip of a ray touching an infinimaximal square frame centered at the origin. A vertex v is called a standard vertex if its embedding is a standard point and non-standard if its embedding is a non-standard point. By the straightline embedding of their source and target vertices, edges correspond to either affine segments, rays or lines or are part of the bounding frame.

Figure 16.2:  Extended geometry: standard vertices are marked by S, non-standard vertices are marked by N. A: The possible embeddings of edges: an affine segment s1, an affine ray s2, an affine line s3. B: A plane map embedded by extended geometry: note that the frame is arbitrarily large, the 6 vertices on the frame are at infinity, the two faces represent a geometrically unbounded area, however they are topologically closed by the frame edges. No standard point can be placed outside the frame.

Extended geometry

Inherits From



the point type of finite vertices.

the ray type of vertices on the frame.

Iterators, handles, and circulators are inherited from Topological_explorer.


Explorer is copy constructable and assignable. An object can be obtained via the Nef_polyhedron_2::explorer() method of Nef_polyhedron_2.


bool E.is_standard ( Vertex_const_handle v)
returns true iff v's position is a standard point.

Point E.point ( Vertex_const_handle v) returns the standard point that is the embedding of v.
Precondition: E.is_standard(v).

Ray E.ray ( Vertex_const_handle v) returns the ray defining the non-standard point on the frame.
Precondition: !E.is_standard(v).

bool E.is_frame_edge ( Halfedge_const_handle e)
returns true iff e is part of the infinimaximal frame.